Question

If `f(x)= cos [(pi^(2))/(2)] x + sin[(-pi^(2))/(2)]x,[x]` denoting the greatest integer function,then
A. `f(0) = 0`
B. `f((pi)/(3)) = sqrt((3)-1)/(2)`
C. `f((pi)/(2)) = -1`
D. `f(pi) = 0`

Answer 1

Correct Answer - B
We have , `pi ^(2) ~= 9.8696
`therefore [(pi^(2))/(2)] = 4 and [(-pi^(2))/(2)] = -5`
`therefore f(x) = cos 4x sin (-5x) = cos 4 x -sin 5x `
Thus, we have,
`f(0) =1`
`f((pi)/(3)) = cos.(4pi)/(3)-sin .(-5x)/(3) = - (1)/(2) +(sqrt(3))/(2)= (sqrt(3)-1)/(2)`
`f((pi)/(2)) = cos 2 pi - sin.(5pi)/(2) = 1-1=0`
and ,`f(pi)= cos 4 pi - sin5 pi = 1`